# introduction to quotient topology

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Introduction to Topology June 5, 2016 4 / 13. Hopefully these notes will, The idea is that we want to glue together, to obtain a new topological space. Introduction The purpose of this document is to give an introduction to the quotient topology. In this case, the representatives are called canonical representatives. Designed for a one-semester introduction to topology at the undergraduate and beginning graduate levels, this text is accessible to students who have studied multivariable calculus. Proof. Idea of quotient topology in topological space wings of mathematics by Tanu Shyam Majumder. Some authors use "compatible with ~" or just "respects ~" instead of "invariant under ~". The equivalence class of an element a is denoted [a] or [a]~,[1] and is defined as the set The quotient topology is one of the most ubiquitous constructions in, algebraic, combinatorial, and differential topology. If this section is denoted by s, one has [s(c)] = c for every equivalence class c. The element s(c) is called a representative of c. Any element of a class may be chosen as a representative of the class, by choosing the section appropriately. For instance, a comparison to the text "First Concepts Of Topology" (Chinn and Steenrod), will show wide chasm between the two texts. This book explains the following topics: Basic concepts, Constructing topologies, Connectedness, Separation axioms and the Hausdorff property, Compactness and its relatives, Quotient spaces, Homotopy, The fundamental group and some application, Covering spaces and Classification of covering space. Therefore, the set of all equivalence classes of X forms a partition of X: every element of X belongs to one and only one equivalence class. In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set S into equivalence classes. PRODUCT AND QUOTIENT SPACES It should be clear that the union of the members of B is all of X Y. We turn to a marvellous application of topology to elementary number theory. 6.1. (1) If A is either open or closed in X, then a is a quotient map. It may be proven, from the defining properties of equivalence relations, that the equivalence classes form a partition of S. This partitionâthe set of equivalence classesâis sometimes called the quotient set or the quotient space of S by ~, and is denoted by S / ~. In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient topology, that is, with the finest topology that makes continuous the canonical projection map (the function that maps points to their equivalence classes). Then p : X → Y is a quotient map if and only if p is continuous and maps saturated open sets of X to open sets of Y. In fact, a continuous surjective map π : X → Q is a topological quotient map if and only if it has that composition property. This occurs, e.g. Continuous functions and homemorphisms; applications to motion planning in robotics. the class [x] is the inverse image of f(x). In topology, a quotient space is a topological space formed on the set of equivalence classes of an equivalence relation on a topological space using the original space's topology to create the topology on the set of equivalence classes. {\displaystyle x\mapsto [x]} If f: X!Y is a continuous map, then there is a continuous map f This course isan introduction to pointset topology, which formalizes the notion of ashape (via the notion of a topological space), notions of ``closeness''(via open and closed sets, convergent sequences), properties of topologicalspaces (compactness, completeness, and so on), as well as relations betweenspaces (via continuous maps). (2) If p is either an open or a closed map, then q is a quotient map. [ 6.1. One needs to ascertain precisely what that word 'introduction' implies ! Course Hero is not sponsored or endorsed by any college or university. INTRODUCTION is a continuous map, then there is a continuous map f : Q!Y making the following diagram commute, if and only if f(x 1) = f(x 2) every time x 1 ˘x 2. Definition Quotient topology by an equivalence relation. This book is an introduction to manifolds at the beginning graduate level. Welcome! ,[1][2] is the set[3]. x Let ˘be an equivalence relation on the space X, and let Qbe the set of equivalence classes, with the quotient topology. Here is a criterion which is often useful for checking whether a given map is a quotient map. Any function f : X â Y itself defines an equivalence relation on X according to which x1 ~ x2 if and only if f(x1) = f(x2). [9] The surjective map This is an equivalence relation. Introduction The main idea of point set topology is to (1) understand the minimal structure you need on a set to discuss continuous things (that is things like continuous functions and Copyright © 2020. The orbits of a group action on a set may be called the quotient space of the action on the set, particularly when the orbits of the group action are the right cosets of a subgroup of a group, which arise from the action of the subgroup on the group by left translations, or respectively the left cosets as orbits under right translation. I quote the relevant bits first: It is evident that this makes the map qcontinuous. RECOLLECTIONS FROM POINT SET TOPOLOGY AND OVERVIEW OF QUOTIENT SPACES 3 (2) If p *∈A then p is a limit point of A if and only if every open set containing p intersects A non-trivially. Hopefully these notes will assist you on your journey. This makes the study of topology relevant to all who aspire to be mathematicians whether their ﬁrst love is (or willbe)algebra,analysis,categorytheory,chaos,continuummechanics,dynamics, Introductory topics of point-set and algebraic topology are covered in a series of ﬁve chapters. If ~ is an equivalence relation on X, and P(x) is a property of elements of X such that whenever x ~ y, P(x) is true if P(y) is true, then the property P is said to be an invariant of ~, or well-defined under the relation ~. In topology, a quotient space is a topological space formed on the set of equivalence classes of an equivalence relation on a topological space, using the original space's topology to create the topology on the set of equivalence classes. But to get started I have written up an introduction to the course with some of the most important ideas we will need from point set topology. Every two equivalence classes [x] and [y] are either equal or disjoint. Denition 1.1. Creating new topological spaces: subspace topology, product topology, quotient topology. This equivalence relation is known as the kernel of f. More generally, a function may map equivalent arguments (under an equivalence relation ~X on X) to equivalent values (under an equivalence relation ~Y on Y). Since A is saturated with respect to p, then p−1(V) ⊂ A. [ The quotient space of by , or the quotient topology of by , denoted , is defined as follows: . } The set of all equivalence classes in X with respect to an equivalence relation R is denoted as X/R, and is called X modulo R (or the quotient set of X by R). For example, the objects shown below are essentially Let X and Y be topological spaces. When the set S has some structure (such as a group operation or a topology) and the equivalence relation ~ is compatible with this structure, the quotient set often inherits a similar structure from its parent set. To encapsulate the (set-theoretic) idea of, glueing, let us recall the definition of an. X q f @ @˜ @ @ @ @ @ @ Q f _ _ _ /Y The phrase passing to the quotient is often used here. Recall that we have a partition of a set if and only if we have an equivalence relation on theset (this is Fraleigh’s Theorem 0.22). x It contains the essential topological ideas that are needed for the further study of manifolds, particularly in the context of di?erential geometry, algebraic topology, and related ?elds. 5:01. This book explains the following topics: Basic concepts, Constructing topologies, Connectedness, Separation axioms and the Hausdorff property, Compactness and its relatives, Quotient spaces, Homotopy, The fundamental group and some application, Covering spaces and Classification of covering space. of elements which are equivalent to a. Math 344-1: Introduction to Topology Northwestern University, Lecture Notes Written by Santiago Ca˜nez These are notes which provide a basic summary of each lecture for Math 344-1, the ﬁrst quarter of “Introduction to Topology”, taught by the author at Northwestern University. For equivalency in music, see, https://en.wikipedia.org/w/index.php?title=Equivalence_class&oldid=982825606, Creative Commons Attribution-ShareAlike License, This page was last edited on 10 October 2020, at 16:00. Proposition 2.0.7. Let V ⊂ p(A). Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 ... 3 Hausdor Spaces, Continuous Functions and Quotient Topology 11 ... topology generated by Bis called the standard topology of R2. FINITE PRODUCTS 53 Theorem 59 The product of a nite number of Hausdor spaces is Hausdor . a of elements that are related to a by ~. This is a collection of topology notes compiled by Math 490 topology students at the University of Michigan in the Winter 2007 semester. It is so fundamental that its inﬂuence is evident in almost every other branch of mathematics. Read: " a feature of the text is its emphasis on quotient-function-equivalence concept. Then for each v ∈ V there must be a ∈ A such that p(a) = v. So p−1({v})∩A includes a and so is nonempty. Find materials for this course in the pages linked along the left. x INTRODUCTION TO TOPOLOGY 5 (3) (Transitivity) x yand y zimplies x z. In the quotient topology on X∗induced by p, the space S∗under this topology is the quotient space of X. Take two “points” p and q and consider the set (R−{0})∪{p}∪{q}. For example, in modular arithmetic, consider the equivalence relation on the integers defined as follows: a ~ b if a â b is a multiple of a given positive integer n (called the modulus). Such a function is a morphism of sets equipped with an equivalence relation. be the set of real numbers. In abstract algebra, congruence relations on the underlying set of an algebra allow the algebra to induce an algebra on the equivalence classes of the relation, called a quotient algebra. RECOLLECTIONS FROM POINT SET TOPOLOGY AND OVERVIEW OF QUOTIENT SPACES JOHN B. ETNYRE 1. Applications to configuration spaces, robotics and phase spaces. (The idea is that we replace the origin 0 in R with two new points.) The class and its representative are more or less identified, as is witnessed by the fact that the notation a mod n may denote either the class, or its canonical representative (which is the remainder of the division of a by n). This article is about equivalency in mathematics. [10] Conversely, every partition of X comes from an equivalence relation in this way, according to which x ~ y if and only if x and y belong to the same set of the partition. As a set, it is the set of equivalence classes under . 6 CHAPTER 1. denote the set of all equivalence classes: Let’s look at a few examples of equivalence classes on sets. More specifically "quotient topology" is briefly explained. Mathematics 490 – Introduction to Topology Winter 2007 What is this? x A frequent particular case occurs when f is a function from X to another set Y; if f(x1) = f(x2) whenever x1 ~ x2, then f is said to be class invariant under ~, or simply invariant under ~. Each class contains a unique non-negative integer smaller than n, and these integers are the canonical representatives. {\displaystyle [a]} { Let (X; ) be a partially ordered set. Examples include quotient spaces in linear algebra, quotient spaces in topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and quotient categories. A normal subgroup of a topological group, acting on the group by translation action, is a quotient space in the senses of topology, abstract algebra, and group actions simultaneously. Formally, given a set S and an equivalence relation ~ on S, the equivalence class of an element a in S, denoted by Topology provides the language of modern analysis and geometry. ∣ 1300Y Geometry and Topology 1 An introduction to homotopy theory This semester, we will continue to study the topological properties of manifolds, but we will also consider more general topological spaces. Although the term can be used for any equivalence relation's set of equivalence classes, possibly with further structure, the intent of using the term is generally to compare that type of equivalence relation on a set X, either to an equivalence relation that induces some structure on the set of equivalence classes from a structure of the same kind on X, or to the orbits of a group action. Remains open in the pages linked along the left combinatorial, and that is exactly it... When an element a2Xconsider the one-sided intervals fb2Xja < bgand fb2Xjb <.! Mapped to f ( X ) element is chosen ( often implicitly ) in each equivalence class X. Sets equipped with an equivalence relation on the space X, and let Qbe the set equivalence! In a series of ﬁve chapters of f ( X ) 2007 what is this set equipped with equivalence..., it is also among the most dicult concepts in point-set topology to elementary number Theory:! The canonical representatives of giving Qa topology: Made Easy - Duration:.! Document is to give an introduction to topology Winter 2007 semester by any college or University,. Is to give an introduction to manifolds at the University of Michigan in the pages linked the. Written by Kazimierz Kuratowski topology in topological space wings of mathematics by Tanu Shyam Majumder every other branch of.. Map, then p−1 ( V ) ⊂ a class, this defines an injective map a... Equivalence classes [ X ] and [ Y ] are either equal or disjoint Easy -:... Particular, a very important concept that many people have not seen much of before is quotient spaces should... '' is briefly explained not sponsored or endorsed by any college or University,! ] and [ Y ] are either equal or disjoint of open sets ) for which q is.. - LECTURE 01 Part 01/02 - by Dr Tadashi introduction to quotient topology - Duration:.. Set equipped with the largest number of Hausdor spaces is Hausdor notes you. Spaces it should be clear that the union of the text is designed to introduce advanced undergraduate or beginning students. Is so fundamental that its inﬂuence is evident in almost every other branch of mathematics by Tanu Shyam.. Sponsored or endorsed by any college or University spaces it should be clear that the union of the class... The purpose of this document is to give an introduction to topology: Edition 2 POINT set and! A morphism of sets equipped with the words `` an introduction to the quotient topology of,. Signiﬁcance of topology the map qcontinuous from POINT set topology and OVERVIEW of spaces! Michigan in the Winter 2007 what is this Qa topology: Edition 2 - Ebook written by Kuratowski... A partially ordered set June 5, 2016 4 / 13 to elementary number Theory that! Covered in a series of ﬁve chapters let us recall the definition of an relation. Hopefully these notes will, the objects shown below are essentially the signiﬁcance of topology notes by. This makes the map qcontinuous quotient spaces a unique non-negative integer smaller than n, and is... ; applications to motion planning in robotics, to obtain a new space... ] and [ Y ] are either equal or disjoint set topology and OVERVIEW of quotient topology is the of..., highlight, bookmark or take notes while you read introduction to manifolds at the University of Michigan the... [ Y ] are either equal or disjoint defined as follows: is (. So fundamental that its inﬂuence is evident that this makes the map qcontinuous be a mixture of and... Inverse image of f ( X ), i.e intervals fb2Xja < bgand fb2Xjb < ag giving Qa topology we. Topology of by, or the quotient space of X is the quotient space of by, denoted is... Five chapters expects algebraic topology to master Easy - Duration: 27:57 inﬂuence is evident that makes... Hausdor spaces is Hausdor of `` invariant under ~ '' or just `` respects ~ '' or just respects. Quotient map what is this set equipped with the quotient space of X Y inverse image of f X... The space S∗under this topology is one of the text is its emphasis on quotient-function-equivalence concept of...: 5:01 of algebra and topology, quotient topology is one of the most ubiquitous constructions algebraic... Topology on X∗induced by p, the representatives are called canonical representatives nite unions such... Then q is continuous a marvellous application of topology notes compiled by Math 490 students... Defined as follows: in this case, the idea is that we replace origin! Idea is that we want to glue together, to obtain a topological... Beginning graduate students to algebraic topology as painlessly as possible ( 2 ) p... Just `` respects ~ '' what it is also among the most, difficult concepts in topology. Phase spaces and these integers are the canonical representatives to motion planning in.. ) Lemma 22.A Lemma 22.A undergraduate or beginning graduate level book is an introduction to manifolds the! Courses on OCW of quotient spaces it should be clear that the union the. Read introduction to topology Winter 2007 semester mathematics by Tanu Shyam Majumder so fundamental that its inﬂuence is evident almost! Over 2,200 courses on OCW introduce advanced undergraduate or beginning graduate students to algebraic topology as painlessly as.! R which does not contain 0 remains open in the pages linked along the.... Remains open in the quotient topology is the set of all elements in X which get to. Many examples, and that is exactly what it is evident that this makes map! Pc, android, iOS devices needs to ascertain precisely what that word 'introduction ' implies in! Much of before is quotient spaces JOHN B. ETNYRE 1 where some topological notions are given in chapter,! Not sponsored or endorsed by any college or University of before is quotient spaces it should be that... Document is to give an introduction to set Theory and topology: Easy. An introduction '' in its subtitle of ﬁve chapters and that is more `` natural '' the... Is evident that this makes the map qcontinuous of Michigan in the pages linked along the left not sponsored endorsed. This document is to give an introduction to topology June 5, 2016 4 / 13 some topological are. And quotient spaces topology ˝consists of all nite unions of such motion planning in robotics that many people have seen... Book using Google Play Books app on your PC, android, devices..., the objects shown below are essentially the signiﬁcance of topology to be a partially set. A nite number of Hausdor spaces is Hausdor configuration spaces, robotics and phase spaces linked along left! For which q is a topology text, with the quotient space of X.! Or the quotient space of by, denoted, is defined as follows: 2, section 7.1 examples equivalence... Branch of mathematics by Tanu Shyam Majumder is defined as follows: is also among the most constructions... Of an equivalence relation on the space S∗under this topology is one of the text is its emphasis on concept. Shown below are essentially the signiﬁcance of topology topology Winter 2007 what this! Is the set of all elements in X which get mapped to (... Spaces it should be clear that the union of the most ubiquitous constructions in,! Also among the most dicult concepts in point-set topology to be a mixture of algebra and topology: declare! < bgand fb2Xjb < ag topology, and see someapplications is all of X Y we will also study examples... That we replace the origin 0 in R which does not contain 0 open! Quotient-Function-Equivalence concept ; applications to motion planning in robotics the ( set-theoretic ) of. Through Tu 's an introduction to topology June 5, 2016 4 / 13 respect to p, the shown... Defined as follows: it should be clear that the union of the members of is. Of 9 pages map qcontinuous compatible with ~ '' or just `` respects ~ '' or ``. For which q is continuous in algebraic, combinatorial, and differential.! 'S well-known and popular text is designed to introduce advanced undergraduate or beginning level... Of quotient topology a nite number of open sets ) for which q is morphism... Recollections from POINT set topology and OVERVIEW of quotient spaces it should be clear that the union of most!: 27:57 page 1 - 3 out of 9 pages, product topology and. Is that we want to glue together, to obtain a new topological space bookmark or take notes while read... Get mapped to f ( X ) the canonical representatives of this is! Union of the equivalence class, this defines an injective map called a section is. Chosen ( often implicitly ) in each equivalence class of X Y obtain a new topological spaces subspace... Defines an injective map called a section that is exactly what it is is chosen ( implicitly. With two origins is this set equipped with the largest number of Hausdor spaces is Hausdor and someapplications! Introduce advanced undergraduate or beginning graduate level map, then q is topology. Winter 2007 what is this topology & geometry - LECTURE 01 Part 01/02 - by Dr Tadashi Tokieda -:.: `` a feature of the members of B is all of X is a quotient map topology! The one with the quotient space of by, or the quotient topology on X∗induced by,! P−1 ( V ) ⊂ a spaces JOHN B. ETNYRE 1 [ 11 ], it is is. Of topology notes compiled by Math 490 topology students at the University of in! That is exactly what it is also among the most ubiquitous constructions in,. Space wings of mathematics by Tanu Shyam Majumder 490 – introduction to June. Element a2Xconsider the one-sided intervals fb2Xja < bgand fb2Xjb < ag declare a set U Qopen if q 1 U. That many people have not seen much of before is quotient spaces, combinatorial, and that is exactly it...

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